Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials

Wenqing Sun; Lei Chen; Wulan Tuya; Yong He; Rihong Zhu

doi:10.1364/JOSAA.30.002539

Journal of the Optical Society of America A
Vol. 30,
Issue 12,
pp. 2539-2546
(2013)
•https://doi.org/10.1364/JOSAA.30.002539

Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials

Wenqing Sun, Lei Chen, Wulan Tuya, Yong He, and Rihong Zhu

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Wenqing Sun, Lei Chen, Wulan Tuya, Yong He, and Rihong Zhu, "Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials," J. Opt. Soc. Am. A 30, 2539-2546 (2013)

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- Instrumentation, Measurement, and Metrology
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About this Article
History
- Original Manuscript: August 29, 2013
- Revised Manuscript: October 21, 2013
- Manuscript Accepted: October 21, 2013
- Published: November 13, 2013

Abstract

Chebyshev and Legendre polynomials are frequently used in rectangular pupils for wavefront approximation. Ideally, the dataset completely fits with the polynomial basis, which provides the full-pupil approximation coefficients and the corresponding geometric aberrations. However, if there are horizontal translation and scaling, the terms in the original polynomials will become the linear combinations of the coefficients of the other terms. This paper introduces analytical expressions for two typical situations after translation and scaling. With a small translation, first-order Taylor expansion could be used to simplify the computation. Several representative terms could be selected as inputs to compute the coefficient changes before and after translation and scaling. Results show that the outcomes of the analytical solutions and the approximated values under discrete sampling are consistent. With the computation of a group of randomly generated coefficients, we contrasted the changes under different translation and scaling conditions. The larger ratios correlate the larger deviation from the approximated values to the original ones. Finally, we analyzed the peak-to-valley (PV) and root mean square (RMS) deviations from the uses of the first-order approximation and the direct expansion under different translation values. The results show that when the translation is less than 4%, the most deviated 5th term in the first-order 1D-Legendre expansion has a PV deviation less than 7% and an RMS deviation less than 2%. The analytical expressions and the computed results under discrete sampling given in this paper for the multiple typical function basis during translation and scaling in the rectangular areas could be applied in wavefront approximation and analysis.

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$n$	$T_{n} (x)$
$T_{1} (x)$	1
$T_{2} (x)$	$x$
$T_{3} (x)$	$2 x^{2} - 1$
$T_{4} (x)$	$4 x^{3} - 3 x$
$T_{5} (x)$	$8 x^{4} - 8 x^{2} + 1$

$C_{i} (x, y) = T_{m} (x) T_{n} (y)$	Corresponding Aberration	Expression after the Translation $C_{i} + (u \cdot (\partial F / \partial x) + v \cdot (\partial F / \partial y))$	Aberration Changes
$C_{1} (x, y) = T_{1} (x) T_{1} (y)$	Piston	$C_{1}$	Piston
$C_{2} (x, y) = T_{2} (x) T_{1} (y)$	x-tilt	$C_{2} + 2 u C_{1}$	x-tilt + piston
$C_{3} (x, y) = T_{1} (x) T_{2} (y)$	y-tilt	$C_{3} + 2 v C_{1}$	y-tilt + piston
$C_{4} (x, y) = T_{3} (x) T_{1} (y)$	x-astigmatism	$C_{4} + 4 u C_{2}$	x-astigmatism + x-tilt
$C_{5} (x, y) = T_{2} (x) T_{2} (y)$	45 deg astigmatism	$C_{5} + 2 v C_{2} + 2 u C_{3}$	45 deg astigmatism + x-tilt + y-tilt
$C_{6} (x, y) = T_{1} (x) T_{3} (y)$	y-astigmatism	$C_{6} + 4 v C_{3}$	y-astigmatism + y-tilt
$C_{7} (x, y) = T_{4} (x) T_{1} (y)$	x-primary coma	$C_{7} + 6 u C_{1} + 6 u C_{4}$	x-primary coma + piston + x-astigmatism
$C_{8} (x, y) = T_{3} (x) T_{2} (y)$		$C_{8} + 2 v C_{4} + 4 u C_{5}$
$C_{9} (x, y) = T_{2} (x) T_{3} (y)$		$C_{9} + 4 v C_{5} + 2 u C_{6}$
$C_{10} (x, y) = T_{1} (x) T_{4} (y)$	y-primary coma	$C_{10} + 6 v C_{1} + 6 v C_{6}$	y-primary coma + piston + y-astigmatism
$C_{11} (x, y) = T_{5} (x) T_{1} (y)$	x-primary spherical	$C_{11} + 8 u C_{2} + 8 u C_{7}$	x-primary spherical + x-tilt + x-primary coma
$C_{12} (x, y) = T_{4} (x) T_{2} (y)$		$C_{12} + 6 u C_{3} + 2 v C_{7} + 6 u C_{8}$
$C_{13} (x, y) = T_{3} (x) T_{3} (y)$		$C_{13} + 4 v C_{8} + 4 u C_{9}$
$C_{14} (x, y) = T_{2} (x) T_{4} (y)$		$C_{14} + 6 v C_{2} + 6 v C_{9} + 2 u C_{10}$
$C_{15} (x, y) = T_{1} (x) T_{5} (y)$	y-primary spherical	$C_{15} + 8 v C_{3} + 8 v C_{10}$	y-primary spherical + y-tilt + y-primary coma

$C_{j} (x, y) = T_{m} (x) T_{n} (y)$	Corresponding Aberration	Expression after the Scaling	Aberration Changes
$C_{1} (x, y) = T_{1} (x) T_{1} (y)$	Piston	$C_{1}$	Piston
$C_{2} (x, y) = T_{2} (x) T_{1} (y)$	x-tilt	$p C_{2}$	x-tilt
$C_{3} (x, y) = T_{1} (x) T_{2} (y)$	y-tilt	$q C_{3}$	y-tilt
$C_{4} (x, y) = T_{3} (x) T_{1} (y)$	x-astigmatism	$p^{2} C_{4} + 2 (- 1 + p^{2}) C_{1}$	x-astigmatism + piston
$C_{5} (x, y) = T_{2} (x) T_{2} (y)$	45 deg astigmatism	$p q C_{5}$	45 deg astigmatism
$C_{6} (x, y) = T_{1} (x) T_{3} (y)$	y-astigmatism	$q^{2} C_{6} + 2 (- 1 + q^{2}) C_{1}$	y- astigmatism + piston
$C_{7} (x, y) = T_{4} (x) T_{1} (y)$	x-primary coma	$p^{3} C_{7} + 3 (- p + p^{3}) C_{2}$	x-primary coma + x-tilt
$C_{8} (x, y) = T_{3} (x) T_{2} (y)$		$p^{2} q C_{8} + 2 (- 1 + p^{2}) q C_{3}$
$C_{9} (x, y) = T_{2} (x) T_{3} (y)$		$p q^{2} C_{9} + 2 p (- 1 + q^{2}) C_{2}$
$C_{10} (x, y) = T_{1} (x) T_{4} (y)$	y-primary coma	$q^{3} C_{10} + 3 (- q + q^{3}) C_{3}$	y-primary coma + y-tilt
$C_{11} (x, y) = T_{5} (x) T_{1} (y)$	x-primary spherical	$p^{4} C_{11} + (2 - 8 p^{2} + 6 p^{4}) C_{1} + 4 p^{2} (- 1 + p^{2}) C_{4}$	x-primary spherical + x-astigmatism + piston
$C_{12} (x, y) = T_{4} (x) T_{2} (y)$		$p^{3} q C_{12} + 3 p q (- 1 + p^{2}) C_{5}$
$C_{13} (x, y) = T_{3} (x) T_{3} (y)$		$p^{2} q^{2} C_{13} + 4 (p^{2} - 1) (q^{2} - 1) C_{1} + 2 p^{2} (q^{2} - 1) C_{4} + 2 q^{2} (p^{2} - 1) C_{6}$
$C_{14} (x, y) = T_{2} (x) T_{4} (y)$		$p q^{3} C_{14} + 3 p q (- 1 + q^{2}) C_{5}$
$C_{15} (x, y) = T_{1} (x) T_{5} (y)$	y-primary spherical	$q^{4} C_{15} + (2 - 8 q^{2} + 6 q^{4}) C_{1} + 4 q^{2} (- 1 + q^{2}) C_{6}$	y-primary spherical + y-astigmatism + piston

$n$	$L_{n} (x)$
1	1
2	$x$
3	$(1 / 2) (3 x^{2} - 1)$
4	$(1 / 2) (5 x^{3} - 3 x)$
5	$(1 / 8) (35 x^{4} - 30 x^{2} + 3)$

$Q_{j} (x, y) = L_{p} (x) L_{q} (y)$	Corresponding Aberration	Expressions after the Translation of $(u, v)$ $Q_{i} + (u \cdot (\partial Q_{j} / \partial x) + v \cdot (\partial Q_{j} / \partial y))$	Aberration Changes
$Q_{1} (x, y) = L_{1} (x) L_{1} (y)$	Piston	$Q_{1}$	Piston
$Q_{2} (x, y) = L_{2} (x) L_{1} (y)$	x-tilt	$Q_{2} + 3 u Q_{1}$	x-tilt + piston
$Q_{3} (x, y) = L_{1} (x) L_{2} (y)$	y-tilt	$Q_{3} + 3 v Q_{1}$	y-tilt + piston
$Q_{4} (x, y) = L_{3} (x) L_{1} (y)$	x-defocus	$Q_{4} + 5 u Q_{2}$	x-defocus + x-tilt
$Q_{5} (x, y) = L_{2} (x) L_{2} (y)$		$Q_{5} + 3 v Q_{2} + 3 u Q_{3}$
$Q_{6} (x, y) = L_{1} (x) L_{3} (y)$	y-defocus	$Q_{6} + 5 v Q_{3}$	y-defocus + y-tilt
$Q_{7} (x, y) = L_{4} (x) L_{1} (y)$	x-primary coma	$Q_{7} + 7 u Q_{1} + 7 u Q_{4}$	x-primary coma + piston + x-defocus
$Q_{8} (x, y) = L_{3} (x) L_{2} (y)$		$Q_{8} + 3 v Q_{4} + 5 u Q_{5}$
$Q_{9} (x, y) = L_{2} (x) L_{3} (y)$		$Q_{9} + 5 v Q_{5} + 3 u Q_{6}$
$Q_{10} (x, y) = L_{1} (x) L_{4} (y)$	y-primary coma	$Q_{10} + 7 v Q_{1} + 7 v Q_{6}$	y-primary coma + piston + y-defocus
$Q_{11} (x, y) = L_{5} (x) L_{1} (y)$	x-primary spherical	$Q_{11} + 9 u Q_{2} + 9 u Q_{7}$	x-primary spherical + x-primary coma
$Q_{12} (x, y) = L_{4} (x) L_{2} (y)$		$Q_{12} + 7 u Q_{3} + 3 v Q_{7} + 7 u Q_{8}$
$Q_{13} (x, y) = L_{3} (x) L_{3} (y)$		$Q_{13} + 5 v Q_{8} + 5 u Q_{9}$
$Q_{14} (x, y) = L_{2} (x) L_{4} (y)$		$Q_{14} + 7 v Q_{2} + 7 v Q_{9} + 3 u Q_{10}$
$Q_{15} (x, y) = L_{1} (x) L_{5} (y)$	y-primary spherical	$Q_{15} + 9 v Q_{3} + 9 v Q_{10}$	y-primary spherical + y-primary coma

Order	Chebyshev Polynomials	Legendre Polynomials
5	$C_{5} + 0.0078 C_{2} + 0.0078 C_{3}$	$Q_{5} + 0.0117 Q_{2} + 0.0117 Q_{3}$
6	$C_{6} + 0.0157 C_{3}$	$Q_{6} + 0.0196 Q_{3}$
9	$C_{9} + 0.0157 C_{5} + 0.0078 C_{6}$	$Q_{9} + 0.0196 Q_{5} + 0.0117 Q_{6}$
10	$C_{10} + 0.0117 C_{1} + 0.0117 C_{6}$	$Q_{10} + 0.0274 Q_{1} + 0.0274 Q_{6}$
13	$C_{13} + 0.0157 C_{8} + 0.0157 C_{9}$	$Q_{13} + 0.0196 Q_{8} + 0.0196 Q_{9}$
14	$C_{14} + 0.0235 C_{2} + 0.0235 C_{9} + 0.0078 C_{10}$	$Q_{14} + 0.0274 Q_{2} + 0.0274 Q_{9} + 0.0117 Q_{10}$
15	$C_{15} + 0.0313 C_{3} + 0.0313 C_{10}$	$Q_{15} + 0.0352 Q_{3} + 0.0352 Q_{10}$

Order	Chebyshev Polynomials	Legendre Polynomials
5	$1.0039 C_{5}$	$1.0039 Q_{5}$
6	$0.0078 C_{1} + 1.0039 C_{6}$	$0.0098 Q_{1} + 1.0039 Q_{6}$
9	$0.0078 C_{2} + 1.0059 C_{9}$	$0.0098 Q_{2} + 1.0059 Q_{9}$
10	$0.0118 C_{3} + 1.0059 C_{10}$	$0.0137 Q_{3} + 1.0059 Q_{10}$
13	$0.0079 C_{4} + 0.0079 C_{6} + 1.0078 C_{13}$	$0.0098 Q_{4} + 0.0098 Q_{6} + 1.0078 Q_{13}$
14	$0.0118 C_{5} + 1.0078 C_{14}$	$0.0137 Q_{5} + 1.0078 Q_{14}$
15	$0.0157 C_{1} + 0.0157 C_{6} + 1.0078 C_{15}$	$0.0177 Q_{1} + 0.0177 Q_{6} + 1.0078 Q_{12}$

Abstract

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Equations (25)

Journal of the Optical Society of America A